Propeller Selection For Boats and Small Ships

Transcription

1 E Marine Training - Prop Matching - February, Propeller Selection For Boats and Small Ships Chris Barry This course is intended to cover the basic elements of marine propulsion, especially propellers and the selection of standard segmental section, off-the-shelf propellers for boats and small ships. The nominal concentration is on planing and semi-planing boats, but this material is applicable to any sort of boat using commonly available propellers. This course is not about propeller design per se, but about selecting a propeller, essentially from a catalog. There is a great deal of information available about propellers, and we can by no means cover all of it in this course. However, beyond the material presented here, I can answer specific questions in the forum or discussion sections of the course, or point you to other sources,so if you have any particular problems not covered in this handout, please ask, since that is the purpose of having this course on line instead of just reading an article. Propeller matching is sometimes regarded as a black art, but like every other magic trick, it's just a matter of standard methods and practice. The ready availability of computers has made prop calculations easier than ever before. The characteristics of the propellers used on most small boats are relatively easy to calculate, either using a spreadsheet program such as Excel or Lotus 123 or with dedicated propeller software. As part of this course, you have been provided with a series of Excel spreadsheets and a dedicated resistance program to help do this, and part of the course will cover using these tools. The most difficult part of selecting a propeller is predicting resistance, and resistance prediction is also a large subject. However, we will cover the basic aspects of resistance prediction for planing and semiplaning hulls for completeness. Fortunately, in most cases you will be working with an known hull, when you re-power an existing boat, prop up a standard hull, or replace an unsatisfactory propeller. If you are careful about gathering trial data, you can usually get a good resistance data. Getting good trial data and back calculating existing boats will also help you gain experience to deal with new designs.

2 E Marine Training - Prop Matching - February, Propeller Anatomy Efficiency of Propulsion How Props Work Physical Characteristics Hydrodynamic Characteristics Hydrodynamic Equations Cavitation Hull Effects Advanced Prop Design This course first discusses the basic elements of efficiency of propulsion of any type, not only marine propellers but any sort of propulsion other than rockets, since the basic principles apply to any sort of propulsion system that does not carry its own reaction mass. Then the specifics of how a propeller works, will be discussed, and we will look at the physical characteristics of propellers, the hydrodynamic characteristics of propellers, and the equations or formulas needed to calculate specific aspects of propellers to allow us to predict their efficiency, thrust and so on. The actual equations won t be derived in this course, due to the limited time, but some explanation will be give about where they come from and how they work. If you are interested in the derivations of the equations, please refer to the references. Note especially that Saunders, Hydrodynamics of Ship Design, available from the Society of Naval Architects and Marine Engineers, or by interlibrary loan, is a very complete, and highly readable book, (actually three volumes) and though it is expensive, and perhaps a bit dated (written in the 50 s) it presents a very good discussion and background, without too much mathematics, of virtually every aspect of hydrodynamics as might be applied to ships, yachts and even fish. This text probably represents at least a good beginning point for anyone seriously interested in understanding hydrodynamics of marine vehicles. We will then discuss cavitation, the effects of the hull on the propeller and the propeller on the hull that have to be taken into account to select propellers, and then comment shortly on advanced propeller design so that you will have a bit of an idea of what sort of advances are potentially available for special problems.

3 E Marine Training - Prop Matching - February, Propeller Selection Engine - Prop Matching Strategy Computer Programs Resistance Calculation Trial Data Shaft Angle/Strut Drag/Clearance Vibration Nozzles Surface Piercing Props/Jets The basic goal of selecting a propeller is to get the hull, engine and propeller matched to achieve the desired goals in terms of speed, possibly towline pull, thrust and engine loading. This requires some thinking about the overall mission of the boat, and understanding of how the propulsions system should perform overall, not only from an engineering standpoint but from an economic one as well. There are many techniques for matching propellers, most developed prior to the use of computers, but we will discuss only two, which are related, and in fact, will only use one. The older techniques use various graphs and so forth, but the method implemented in the computer spreadsheets given here is the current standard, and probably the most accurate available, at least for standard type propellers. We will discuss resistance calculation in enough detail to understand the problem in general and to use the two methods of resistance given that are specific to many small craft. Most commonly, we will find ourselves working with existing vessels, either to correct problems, or to modify the equipment or service of a boat, such as when it is re-powered. This allows us to get the needed information on resistance by running trials, and then back-calculating resistance. We will discuss the issue of shaft angle, strut drag, and clearance, and a bit about shaft and system vibration, since these often are part of the issue in problem props. Finally, we will discuss the application of nozzles in props and quickly cover surface piercing propellers and water jets.

4 E Marine Training - Prop Matching - February, Efficiency Propulsion Works by Grabbing And Throwing Mass Thrust = m v - but - Energy = 1/2 m v 2 Lowest Thrust per Pound of Water - Best Efficiency Output Velocity Greater than Zero is Wasted Efficiency Depends on Frontal Area Prop Ideal Efficiency Higher as Pitch Gets Higher - More Mass Flows Through - Thrust Load Goes Down Ideal Efficiency Based on Thrust/Speed, Area Device Efficiency - No Device Produces Thrust Perfectly Best Props Typically Produce 80% of Input Power As Water Flow - Best Jets Maybe 90% In general, all propulsion works by throwing mass in the opposite direction you want to go, which, by Newton s law produces and equal and opposite reaction of you moving. The thrust you produce is the mass times the speed you throw it away. (Note that pounds are units of force, not mass - the proper English units of mass are slugs, which incidentally weigh about 32 pounds. A kilogram is also a mass, not a force. The metric unit of force is the Newton. Weight is the force a mass produces due to gravity, which is why it s confusing.) All propulsion works this way; even when walking you are grabbing the earth with your feet and pushing it backwards away from you. However, the earth is so massive compared to you, only you seem to move. All propulsion, except rockets (which carry their reaction mass) also takes in mass from the environment, increases its speed to that of the vehicle, then a bit more, and throws it away. Let's first consider a propeller as just a magic device that grabs and moves water, to see what we can learn without going into detail. To propel boats, we grab water and throw it aft. Thrust is mass times velocity, but the energy required to throw it is one half mass times velocity squared. The more water we grab, the less thrust we need per pound of water so that we can throw it slower with less energy, producing more efficient propulsion. There are then two ways of grabbing a lot of mass, either by moving through the water fast, or by having a large area scoop or whatever, to grab it with. The ratio of thrust to speed, area and water density is expressed as "thrust load coefficient", C tl, which can be proven to determine the maximum or ideal efficiency η i of any propulsor: η i = 2 / ( 1 + (C tl + 1)1/2 with C tl = Thrust / 0.5 ρ A V a 2 ρ, rho, is water density, A is area, V a is the speed of the device through the water. Thrust load coefficient is simply the ratio of thrust to the weight of water that passes through the device per second times half the speed it passes through. Since no real device is perfect at accelerating water, real world efficiency is less that the ideal.

5 E Marine Training - Prop Matching - February, Thrust Load Diagram Thrust Load Is The Main Limit Of Efficiency 100% 90% Ideal A nd Typical Efficiency M aximum Ideal Typical Propeller M ax imum T ypical Real Prop Efficiency 80% P/D 1.2 Efficiency 70% 60% P/D 0.8 P/D % 40% 30% P/D 0.6 Higher Boat Speed or Larger Device Areas = More Mass Flowing through Device Thrust Load Coefficient Ctl = Thrust / 0.5ρAVa 2 This figure shows ideal efficiency plotted against thrust load factor with the actual efficiency of some typical props. The plot also has a constant line at 80% of ideal efficiency, which is the typical limit for propellers. This is sometimes called device efficiency and is the measure of how good a device is at moving water. Incidentally, this figure does not have anything about water in it, and applies equally well to air propellers, but air is much less dense than water, so ρ is small. There is nothing specific about propellers in this plot. It applies equally well to waterjets, oars, paddlewheels or any other forms of fluid propulsion (provided we can identify the area of the device. Some devices may have higher (water jets or some types of sculling hydrofoils - also known as penguin propulsion can be a bit higher) or lower (paddlewheels and oars have about 50%) device efficiencies, but we can calculate their approximate efficiency from this plot This plot shows that thrust load factor is the most important factor for efficiency. The larger the propulsor, (and usually the slower it turns), and the faster it goes through the water, the better the efficiency, though there are other limits on diameter, not only practical limits such as draft, but secondary propeller effects that limit the efficiency of large props. The plot also shows this. If we don t accelerate the water at all, we spend no energy, but get no thrust. This is theoretically very high efficiency, but gets nothing, and since in reality, no device is perfect there is a limit to maximizing the area of the device.

6 E Marine Training - Prop Matching - February, How Props Work A Prop is A Wing Look At A Section Of A Blade: Flow Into Blade: Vector Sum V a + RPM * 2πr (r is the local radius to the section) Wing Generates: Lift (Perpendicular To Flow) And Drag (Parallel To Flow) Lift, Drag Components Resolved Along Shaft Force Along Shaft: Thrust Moment Across Shaft: Torque Face is the Aft (Pressure) Side Torque Drag Flow Due To RPM Thrust Total Force Lift Back is the Fwd (Suction) Side Combined Flow Into Prop Forward Flow Due To Boat Speed V a A propeller is several blades sticking out of a shaft, which rotates and moves forward with the boat. The combination of forward speed and rotation adds up at each section of a blade so that the blade is traveling diagonally across the shaft. Each blade is a wing that produces lift at right angles to the direction of water flowing into the blade and a much smaller drag backwards as shown above. These forces can be resolved as shown to those acting along and across the shaft. Thrust is the total force along the shaft from each section of each blade. The torque that's required to spin the shaft equals the total forces across the shaft from each section, times the radius from the sections to the shaft centerline. Clearly, the angle of flow at each section is important to the relationship between torque and thrust. However, this angle varies from the hub to the tip, so it is easier to characterize the angles by the ratio of speed divided by rotational speed times diameter. This is called the advance ratio, J. J goes up as the boat moves faster relative to the RPM and goes down as shaft RPM goes up relative to boat speed. The actual angle of the blade relative to the shaft is also very important, but this also varies, and so we have a different way of characterizing this angle as well, called pitch. Slip is another sort of advance ratio, but it compares pitch and RPM to speed, instead of diameter and RPM to speed. Since most propeller data is based on pitch to diameter ratio, we could use either form of advance ratio, but standard practice happens to based on diametric advance ratio.

7 E Marine Training - Prop Matching - February, Propeller Characteristics Diameter D - Radius R = 1/2 Diameter Pitch - Stock Props Are Constant Pitch Blade Section Segmental - Cheaper, Better For Cavitation Airfoil Section Area - DAR, EAR, Projected Area Skew - Noise Control, Weedless, Cavitation Rake - Improve Shaft Inclination, Clearance Cup - Reduces Cavitation, Adds ~1 of Pitch The important parameters that characterize a propeller are thus diameter, (and/or radius, half or the diameter), pitch, the type of blade section, the area of the blades, which can be expressed in various ways, the skew angle of the blades, the rake angle of the blades, and in the case of stock segmental props, the leading or trailing edge cup. In this case, we are mainly dealing with constant pitch propellers with segmental sections, which are a slice of a circle. The face (the aft side, which produces pressure) is flat and the back (the forward side which produces suction) is a circular arc. This shape is reasonably efficient, backs down well, resists cavitation reasonably well, and is inexpensive to design and make. Other propellers can have airfoil or wing like sections, cambered (curved) wedges, or other shapes for special purposes. The Troost or Wageningen B series of propellers is commonly used for large ships and has partly airfoil sections, which slightly increase efficiency a bit, but are more susceptible to cavitation. (Note that some methods of propeller matching, using Bp-δ charts, are based on this series. The airfoil section slightly changes the lift and drag produced at a given angle to flow. As a result, these propellers act as if they had a little bit less pitch than the equivalent segmental propeller, so using Bp-δ charts with segmental propellers will tend to overpitch the propeller, and the engine may not make required RPM.) One way to improve the cavitation performance of segmental propellers is to bend down the last inch or so of the trailing edge a few millimeters to form a slight radius of about millimeter. This increases the effective pitch a small amount, and does so in the region of the pro not commonly subject to caviattion (cavitation is gnerally a problem toward the leading edge) and it also creates a pressure change at the trailing edge which tends to suck the cavitation bubble off the back and thus delay cavitation even a little bit more. Many segmental propellers also have a little wedge ground out of the leading and trailing edge. A rounded edge will tend to produce vortices that alternatively form from the face and back. This alternation produces noise; singing. By cutting the edge at a hard angle one way or the other, the vortices don t switch and this anti-singing edge eliminates the noise.

8 E Marine Training - Prop Matching - February, Pitch, Developed, Expanded & Projected Area Pitch: Blade Face Defined by a Helix Pitch is Distance Helix Advances in One Rotation (Tip Travel = 2πR) This Means: Angle Along Blade Face Increases As Sections Get Close to Hub Section Angle Φ At Radius r From Shaft Tan Φ = Pitch / 2πr Propellers Designed As Flat Surface: Expanded Area, A E Propeller Wrapped Around Shaft: Developed Area, A D Propeller View Down Shaft: Projected Area, A P Developed Area Projected Area Back Expanded Area Face Pitch A standard propeller face is part of a helical surface, like the noses of treads on a spiral staircase. A helical surface is formed by a line that advances at a constant speed while it also rotates. Each line is spaced forward (or up in the figure above) uniformly. Each line also is uniformly rotated compared to the next one. This means that the distance along the shaft angle between an inner end of one line and the next inner one is the same, as is the distance between an outer end and the next outer one. However, the distance around the shaft (not the angle, but the distance), is small at the inner end and large at the outer end. Thus the angle between one end decreases as you go out from the shaft centerline, or increases as you go in. There is no single angle that characterizes the surface, but both ends travel the same distance forward, so we can characterize all angles by this distance, which is the pitch. The actual angle at any radius can then be easily calculated from the pitch and the radius. We can also look at the propeller non-dimensionally, which allows us to scale the data from two similarly shaped propellers of different sizes by specifying the pitch to diameter ratio, P/D. Since the angle of the water flow into the section is the sum of the speed of the prop through the water (which is constant with radius) and the speed of the blade around the shaft times the radius, (which increases away from the shaft centerline), the angle that the flow assumes into the sections varies in the same way and each section sees the desired angle of attack of flow into the section. Some ship propellers are designed taking into account the specific flow pattern at the stern, and have a pitch that changes from one radius to another, but this is rare in boats because such props are expensive to design and build. (This is variable pitch. Propellers that can change pitch mechanically are called controllable pitch.) We can also characterize the area of the propeller is various ways: The propeller blade is initially designed as a flat surface. The sections are drawn rotated into the paper. This view is the expanded view, and the area is the expanded area, A E. Wrapping the flat blades around the shaft results in a slight change of area to the developed area A D. This is the actual area of the face in three dimensions. We can also look at the propeller end on and see its projected area, A P.

9 E Marine Training - Prop Matching - February, Standard Propeller Drawing Rake Skew 100%R 80%R 60%R B t80% C t60% D t80% t60% B D Developed A D A Projected 40%R 20%R t40% t20% Hub t40% t20% C Prop CL Elevation (Side) View t Expanded View Pitch/2π Blade Angles Section (End) View Pitch/2π This is the standard layout for a propeller drawing. The drawing starts with the expanded view, which shows the sections and skew, or sweep back angle, of the blades. Skew make the propeller enter a given flow area less suddenly as it spins than if all of the sections were aligned. This reduces noise and change the loading along the blade. Skew can also be used to make weedless propellers. Typical stock propellers are skewed from zero to about fifteen degrees. Faster running propellers tend to need skew more than slow ones. More than about 15 degrees of skew or so greatly increases the bending load on the propeller, so highly skewed propellers are only used on submarines or cruise ships or other vessels where low noise is very important. Since the surface moves the pitch distance forward when the propeller turns through a full circle (2π radians), the pitch angles are laid out by measuring the pitch divided by 2π along the center line and drawing a line from this point to each radius. Each section is laid out along the line, and the distances along (B and C) and across (A) the shaft are determined by projection. The elevation view shows the side view, including the blade thickness and the rake angle. Rake takes advantage of the fact that the flow into the propeller is slightly inwards. It also increases the clearance between the blade an the hull. Raked propellers are not common on small craft and tend to be more expensive to make. The blade thickness reduces away from the shaft center, so the nominal thickness is the thickness projected as if the blade went all the way to the centerline. The aft or trailing edge of the propeller is distance B aft of the face of the section, and the forward or leading edge is distance C forward. The projected view is laid out by wrapping the distance along the shaft (A) in an arc around the shaft centerline. The radius of curvature of a helix is based on the radius at the section, and the pitch and is laid out by the intersection of the section line and a right angle back to the centerline as shown. The developed view is approximated by laying out distance D around this center. (A helicoidal surface is warped, and like any other double curved surface can t actually be exactly laid out flat without some distortion, but this approximation is accurate enough for most propellers. It is less accurate for very wide blades and very high pitch to diameter ratios.)

10 E Marine Training - Prop Matching - February, More Propeller Parameters Z - Number of Blades Disk Area, A 0 = πd 2 /4 = πr 2 Expanded Area Ratio, EAR = A E / A 0 EAR ~ 0.34 * DAR * (2.75 (DAR / Z) Developed Area Ratio, DAR = A D / A 0 Projected Area Ratio, PAR = A P / A 0 Blade Thickness Ratio = t (at CL) / D (see elevation view) Blade Width Ratio = Maximum Blade Width / D Mean Width Ratio = Average Blade Width / D The number of blades is another parameter, and because N is revolution rate, Z is used for this. More blades produce a smoother thrust, increase thrust and reduce cavitation, but tend to reduce efficiency. As discussed above, the various areas characterize the propeller as well. All of these areas can be non-dimensionalized for using at the desired scale by dividing by the propeller disk area. Typical ranges of DAR or EAR are from about 30% to 80% for most propellers, with some two blade props at 30% and some five blade props over 80%. (Manufacturers generally give DAR.) Note that EAR and DAR are two different things, and there is an approximate conversion formula. though they are generally very close to each other numerically. If you actually look at the conversion for typical stock propellers, the difference is only a few percent. Increasing DAR or EAR reduces efficiency a bit, but reduces cavitation a lot, so high powered props generally have high DAR/EAR. These props also have more metal and hence are more expensive. Very high DAR/EAR can be appropriate for some heavily loaded props, but they can get very expensive because the blades overlap and are difficult to cast and machine. The maximum width ratio also characterizes the blade shape, as does the mean (or average) width ratio. Obviously the mean width ratio, times the length of the blade, is the blade area, and that times the number of blades is the expanded area. In terms of appearance, low EAR / DAR blades look like a rabbits ear s, but high EAR/DAR props look like Mickey Mouse s ears. Likewise, the thickness can be non-dimensionalized for using at the desired scale by dividing by the propeller diameter. Propeller quality - tolerance on uniformity of pitch and other dimensions is a final characteristic. Props generally are manufactured to either ISO 484 or an informal but widely used US standard. ISO props come in grade 3, 2, 1 and S with 3 being the worst and S being the best. The informal US standard is approximately equivalent to ISO 2, and generally provides satisfactory performance for boats up to about 25 knots or so.

11 E Marine Training - Prop Matching - February, Prop Hydrodynamic Equations I Thrust Coefficient: K t = Thrust / ρ D 4 n 2 Torque Coefficient: K q = Torque / ρ D 5 n 2 Advance Ratio: J = V a / n D Cavitation Numbers ρ D n V a P t σ 0.7r = P t / ( 1 / 2 ρ (V 2 a + (0.7 π n D) 2 ) τ crit = Thrust / ( 1 / 2 A p (V 2 a + (0.7 π n D) 2 ) Propeller Efficiency (Measured in Open Water) η o =(J/2π) (K t / K q ) A p σ 0.7r τ crit Symbols (English Units) Water Density, Ft-Lbs/s Propeller Diameter, Ft Propeller Rev/Sec Prop Speed of Advance, Ft/s Vapor Pressure At Prop, Lbs/Ft 2 ( * Hub Submergence, Ft) Projected Prop Area, Ft 2 Cavitation 70% r Critical Thrust Coefficient It is possible to calculate the flow over the propeller surface directly with very sophisticated computer programs, but most boat propellers are basically similar, so you can use test data instead. You can test a propeller in a water tunnel at different combinations of water and shaft speeds, measuring torque, thrust and efficiency as a function of advance ratio, J. The "thrust coefficient" (Kt) is the ratio of thrust to shaft speed, diameter and water density. "Torque coefficient" (Kq) is a similar ratio for torque. Both these factors vary with advance ratio and depend on the prop characteristics, but the tests allow us to plot them or fit them to statistical calculations. Once we know the thrust or torque coefficient for a given J, we can then calculate the thrust produced by the prop or the torque required to spin it at a certain speed. (Torque or thrust is just the coefficient times the factors in the denominator.) Cavitation number is based on the ratio of the water pressure at the propeller hub to the characteristic pressure produced by the speed of water. Various speeds are used, but in this case, we will use the speed at the section 70% of the radius away from the shaft centerline, since this is often considered to represent the average conditions on the propeller. Remember that this speed is the sum of the forward speed and the speed due to the revolution of the prop. We can develop another cavitation related measure based on the thrust the propeller is producing divided by the projected area times the characteristic pressure due to speed. Since thrust divide by area is a pressure this will help us to determine limits of thrust based on cavitation. Efficiency is the ratio of thrust and speed to torque, so it is determined by Kt, Kq and J. Efficiency, Kt, and Kq are usually plotted against J for a range of pitches for one style propeller. Note again that theses ratios are non-dimensional. These ratios are used instead of actual RPM, speed, thrust and torque so that they are easily scaled to whatever size and speed prop is required.

12 E Marine Training - Prop Matching - February, J-Kt-Kq Diagram Increasing Shaft RPM Segmental Propeller, 4 Blade, 0.75 EAR Efficiency P/D 1.2 Kt, Kq, Efficiency Kt P/D 0.6 P/D 0.8 P/D 1.0 Increasing Boat Speed 0.1 Kq Advance Ratio, J Two systematic series of propeller tests, the Gawn-Burrill and Newton-Rader systematic series, were run in the Fifties to determine the characteristics of typical boat propellers. This data is widely used to determine propeller coefficients without actually running tests. These tests define propellers by the number of blades (Z), pitch divided by diameter (P/D), and the expanded area ratio (EAR). These propellers all have segmental sections and small skew. This is a standard plot using data from a statistical fit to these series, and these plots were used to match props prior to computer methods. Look at this plot in detail. First, note that both the torque (Kt) and thrust (Kq) coefficient decrease as the advance ratio (J) increases. This is why failure to make speed is a problem. If you selected a prop expecting a certain speed, and got less, the advance ratio is lower and the torque coefficient is higher, so the shaft needs more torque to spin at the rated speed. The plot also shows that the efficiency of a propeller increases steadily as advance ratio increases until efficiency reaches a peak and then falls off radically. The increase of efficiency is due to the increased speed of the propeller through the water compared to the speed of outflow from the prop - reduced thrust load producing a higher ideal efficiency. For the same reason, higher pitch propellers have a higher peak efficiency, occurring at a higher advance ratio. The fall off occurs because the angle of the flow at each section is no longer enough to produce efficient lift. However, for a given advance ratio, the efficiency of a lower pitch prop is better. This is because the blades of a lower pitch prop are twisted more perpendicular to the shaft so that the lift is pointed more along, and less across, the shaft, giving more thrust for less torque. Finally, note that the range of best efficiency is fairly narrow. If you operate a prop at the wrong advance ratio, you will waste a lot of power. You can easily be forced to use an inefficient propeller by a limited prop diameter or a low gear ratio. If you are re-powering a boat, make sure that you can fit in a good prop before you take the job.

13 E Marine Training - Prop Matching - February, Prop Hydrodynamic Equations II Thrust Coefficient (Fit To Test Data) K t = Σ C Ti J si P/D ti EAR ui Z vi Torque Coefficient (Fit To Test Data) K Q = Σ C Qi J si P/D ti EAR ui Z vi Blount - Fox Thrust Criteria: K t / J 2 = R T / ( ρ D 2 v a2 ) - This is a method to find RPM if you know resistance, speed and diameter - the equation eliminates RPM. You calculate curves of efficiency, etc. by K t / J 2 and can then optimize them and pick the point where you get the right K t / J 2, then calculate required RPM from J. Similar equations can be developed to find any other factor. Square of Velocity at 70% r v 0.7r = (J / J 2 )v 2 The obvious question is where to get the thrust and torque coefficients if we can t do propeller tests. There are several sets of data based on statistical analyses of many tests. Each of these sets has been fit to a very messy pair of equations by extensive computer analysis. The equations say that the coefficient is the sum of up to 47 terms. Each term is an arbitrary constant times advance ratio, J, to an arbitrary power, times pitch to diameter ratio, P/D, to another arbitrary power, times EAR to a third arbitrary power, times Z to a fourth arbitrary power. There is a different equation for Kt and Kq, and different tables of constants and powers for different types of propellers. Though these equations are very messy, they don t have to be algebraically manipulated in any way. You only have to plug in your J, P/D, EAR and Z, multiply and add. This would be quite tedious by hand, but it s a snap on a computer, even with a spreadsheet. Incidentally, the equations are a statistical fit to data and don t necessarily make sense on their own, or outside the range of data that was used to develop them. It is interesting to enter nonsense such as one or zero bladed propellers or negative pitch ratios and see what comes out, but this is a reminder that the equations are not valid outside of the data that was used to generate them. Another equation that is frequently useful, though not used in this course, is the Blount-Fox thrust criteria. This is produced by dividing thrust coefficient by J squared and substituting and simplifying. The result is a coefficient that can also be produced by dividing thrust by density, diameter and speed. The merits of this coefficient is that RPM doesn t enter the equation, so that you can determine some propeller factors before you know RPM, then solve for RPM. Blount and Fox produced charts of this factor for typical small craft propellers, to reduce the effort of hand calculations and it remains a useful technique for early design, and solving some specific problems. However, the speed of computer methods allows you to beat the problem to death by just guessing at RPMs repeatedly using the standard equations. Another factor, the square of velocity at 70% radius (which is used in the cavitation equations) can also be determined in terms of J alone (the resulting number is the same). This appears in some methods, and though not used in this course is included for completeness.

14 E Marine Training - Prop Matching - February, Cavitation Boiling Water - Not Air Entrainment High Tip Speed And Loading (Too Much Suction) Causes Thrust & Torque Breakdown, Damage Bubbles Collapse Against Blade, Rudder, Hull, To Prevent Cavitation: Increase Area - More Blades, Bigger Blades (High DAR) Cupping, Blunt Trailing Edge Sucks Bubble Off Blade Cupped Leading Edge Forms A Shock Free Entry Special Sections, Usually With Maximum Thickness Well Aft Supercavitating Props - Form A Stable Bubble A common problem that comes up is cavitation. When the propeller tries to create too much lift for its area, the vacuum on the back (forward) side of the blade is so intense that the water boils. The resulting pockets of steam cause loss of the suction that provides lift, and therefore loss of efficiency, though the prop still produces thrust. Worse, the pockets of steam collapse back into liquid when they reach areas of higher pressure, and the water rushing in to fill the void they leave creates an implosive impact that can damage the propeller or underwater components downstream. You can often hear cavitation by listening to the hull plating near the propeller. It sounds like the boat is running in gravel. Cavitation produces a pitting on the propeller as well, generally on the forward side or face near the leading edge. Sometimes a small portion of the prop at a constant radius will show cavitation damage. This may be due to something in front of the prop changing the flow into it. Look forward for things like scoops or poorly faired struts, and fair them in. The normal standard for cavitation is when no more than ten percent of the back of the blade is covered in vapor bubbles. This limit is represented as another standard equation, developed by Blount and Fox. To avoid cavitation, first reduce the RPM of the prop. This requires higher pitch and perhaps a different gear ratio. Reduced RPM may also require a propeller too large to fit under the boat. In this case, increase the loaded area of the propeller by increasing blade area ratio or the number of blades. Finally, cupping acts like increased pitch, but also acts to suck the cavitation bubble off the back of the blade. You may be able to cup a propeller instead of ordering a new one if you are close in pitch or marginal in cavitation. There is limited data on cupping but MacPherson has some data that suggests that a cupped prop can be assumed to absorb torque and produce thrust like it was cupped but to cavitate as if the cup was not present. The spreadsheet includes this estimate of cup effects - run the case with cup to get thrust and torque and without to determine the level of cavitation. Cavitation is not the same as "ventilation", which is ingestion of air. Ventilation causes reduced efficiency, but not damage. Props also continue to produce thrust when ventilating and surface piercing props are intentionally designed to ventilate. High speed service may require fully cavitating or ventilating props. Special methods have to be used for these conditions.

15 E Marine Training - Prop Matching - February, Standard Cavitation Diagram Plot σ 0.7r Versus Log(τ crit ) and Compare Against Standard Criteria % Back Cavitation Cavitation Criteria 0.7 5% Back Cavitation 10% Back Cavitation Blount-Fox 10% Line % Back Cavitation 30% Back Cavitation 0.5 Cavitation Log(τcrit) σ(0.7r ) The standard criteria for small ship and boat (especially planing boat) cavitation was also developed by Blount and Fox based on the GAWN and Newton Rader data. The light lines show various percentages (in terms of area covered by bubbles) of back cavitation. The two axes are the cavitation number at 70% radius and the log of the critical thrust factor. The dark line is a linear fit to roughly ten percent back cavitation, which is the level at which cavitation is considered to begin to be important. This line is when: τ c = σ 0.7R 0.88 Once τ c exceeds this factor, the propeller will be seriously cavitating, and you may have to take steps to fix the problem. In general, these steps will be to increase the DAR/EAR or to decrease the speed of the blades, which might require increasing the pitch. There are several other cavitation related factors, notably the thrust and torque for fully cavitating conditions. These are coded on the spreadsheets, but are not really worth discussing. Note that this is a standard simple method for evaluating cavitation. More sophisticated methods, including model tests and direct simulation with very sophisticated computer programs are available when appropriate, but these methods are beyond the scope of this course and should be referred to specialists in propeller design. It is also worth noting that there are tables/formulas for performance of cavitating standard propellers, but again, the goal is to avoid cavitating conditions.

16 E Marine Training - Prop Matching - February, Wake Fraction And Thrust Deduction Hull Affects Prop - Prop Affects Hull Wake Fraction - 1-W t : Slower Flow Into Prop Due To Hull 6 0 Shaft: Semi-Planing & Planing 100% - 96%, 12 0 Shaft: Semi-Planing & Planing 97% - 95%, Slow Displacement - Twin Screws: 90%, Single: 80% Outboards, Outdrives: 97% Thrust Deduction - 1-t: Lost Prop Thrust: Hull, Rudder 6 0 Shaft: Semi-Planing & Planing 98%-99%, 12 0 Shaft: Semi-Planing & Planing 98% - 99%, Slow Displacement - Twin Screws: 92%, Single: 85% Outboards, Outdrives: 100% Both Depend On How Much Hull In Front Of Prop, Rudder Most propellers operate behind the hull of a boat, but standard thrust and torque tests are run with the propellers operating in clear water in a tunnel. (The mechanism supporting and turning the propeller is behind it.) You have to correct for the effect of the hull on the propeller. The hull slows down the water entering the prop. The percentage the boat slows down the water is called "wake fraction". Typical planing and semi-planing boats have a wake fraction less than 5%, so a boat may be going 20 knots, but the propeller seems to go a bit over 19 knots. The water near the boat is being dragged along at about one knot. Strictly speaking, there two wake fractions, one for thrust and one for torque, but these two factors are generally quite close, so you can usually assume they are the same. The suction of the propeller also adds drag to the boat hull so it doesn't really produce as much useful thrust as it would in a water tunnel test. This is called "thrust deduction", and is also about 5% for planing boats. Both factors depend on shaft angle. Some hulls also make the flow into the propeller rotate. This makes the propeller acts as if it was spinning slower or faster and is called "relative rotational efficiency". There are even devices that do this intentionally to improve efficiency, especially for large ships, but in most boat hulls, this factor is unity. There is no one simple method of estimating wake fraction and thrust deduction for all hulls but there are estimating techniques and typical data available for most boat types. These factors are best found from trials data. Even if you are not exact, the trial data will tend to cancel out the effects. For a first cut, you can use 10%-20% for wake fraction and thrust deduction for most displacement type hulls like trawler yachts or sailboats with the propeller in a cutout in the keel, 10% for sailboats with fin keels and a prop on a strut and 0%-5% for most planing boats. Multiply the speed by one minus the wake fraction. This is the "velocity of approach", Va, the speed the prop actually sees. It is also worth checking to see how much difference it makes, by checking a range of values.

17 E Marine Training - Prop Matching - February, Hull Efficiency, Relative Rotational Efficiency, Quasi-Propulsive Efficiency Hull Efficiency: η h = η r (1-t) / (1-W t ) Boat Uses Power Based On Boat Speed, V Non-Propelled Resistance, R T (With Appendages, Air) Prop Produces Power Based On V a =(1-w)V Prop Produces Thrust Based On T = R T / (1-t) Relative Rotational Efficiency: η r Due to Swirl Into Prop Typically Small: 101% - 97%, Quasi Propulsive Efficiency: η D = η o η r (1-t) / (1-W t ) η D =EHP / Delivered HP = (R T V / T V a ) * η o η r May or May Not Include Gearset, Shaft Losses (~3%) Since hull efficiency and other terms are frequently used, it s important to understand them. The effect of the propeller in slowing down flow into the propeller acts to reduce the power the propeller puts out because power is force times speed. However, if the propeller has to produce extra thrust (because of thrust deduction) to push the boat through the water, this is a loss. Finally if the flow around the hull acts in some way to change the propeller efficiency there are more potential losses or gains. In one way, these factors might be considered an accounting issue, since all of these factors come from the effort to predict power requirements from model tests run in specific standardized ways. Unfortunately, there is no source for small scale water. You have to use regular water, and its viscosity (stickiness) is proportionately larger than it would be at full scale. Thus effects due to viscosity are larger at small scale, and they have to be corrected for by subtracting out calculated small scale viscous effects (mainly skin friction) and then adding back calculated large scale effects. The result of this is that the force to move a full size vessel that weighs say 100,000 pounds is less than 1000 times that of a model that weighs 100 pounds. Traditionally, initial resistance model tests are run unpowered, and also often without appendages like struts and shafts. Sometimes actual measurements of flow velocities in the propeller area are made during these tests to help determine wake fraction. If a series of self-propelled tests is run, the model is also pulled to make up the additional drag caused by small scale and the scale speed of the propeller is noted. The various thrust deductions, wake fractions and so on are then determined from the various scaling adjustments to this process, and these interaction factors essentially get the books to balance out. Thus we have a hull efficiency that expresses the effect of wake fraction and thrust deduction on the efficiency of propulsion, and then a quasi-propulsive efficiency that takes into account everything except bearing and gear losses (sometimes sometimes they are included make sure you know what is meant - the difference is about 3%). For most boats, this is about 55% of less, so ultimately, you have to buy about twice the engine as the hull alone requires with perfect propulsion.

18 E Marine Training - Prop Matching - February, Advanced Prop Design Use Of Advanced Sections Cavitation Resistance Is Most Common Goal High Lift Sections Adapted From Modern Wings Wake Adapted Adjust Radial Pitch to Match Inflow Improved Cavitation Characteristics, Efficiency Common Strategy for Outboard Engines Circumferential Variations Cause Problems Relative Rotational Efficiency Devices Reaction Fins - Proven on 41 UT Grimm Wheels Though this course is not about advanced prop design, it is worth knowing what can be done by experts with sophisticated computer programs and possibly model tests. The most common problem is to get an effective propeller on a boat that is too fast, or too heavily loaded for the feasible propeller diameter, so that cavitation is a problem. This problem can be attacked through refinements of the section. Barn roof sections are designed to have a suction distribution that comes up to the maximum possible value that can be sustained without cavitation and then remains at this value until the pressure is dropped suddenly near the trailing edge. (The term barn roof refers to this pressure distribution, not the section shape itself. They were originally developed for short take off aircraft which have to delay stalling, a problem somewhat similar to cavitation.) These sections generally have a carefully rounded nose, their maximum thickness well aft and a reverse turn in the back surface. Though these sections are highly resistant to cavitation, they are not as efficient as some other sections. There are also other tricks with skew, variable rake and other things (including holes through the blade) that can help as well. These propellers can look very peculiar. Wake adapted propellers use the fact that the flow into a propeller is generally not uniform, due to the effect of the hull. Unfortunately, a propeller blade has to rotate all the way around the shaft, so it can t specifically adapted to a certain area, but none the less, there is some possible advantage to changing the pitch close to the hub compared to the tips based on the field of flow. This type of adaptation works best in the flow around a big bossing on a single screw, so wake adapted propellers are very common on large ships, but there may occasionally be advantages for wake adaption, particularly on large hubs behind big objects, such as outboards or I/O legs. A final strategy is to take advantage of the rotational flow due to the prop and extract the energy remaining in the swirl. The Volvo Penta DuoProp is just such a device. Small fins that cause the flow into the prop to swirl in the opposite direction also can work sometimes, and a device called a Grimm wheel is an extra unpowered propeller that is spun by the swirl left in by the powered one.

19 E Marine Training - Prop Matching - February, Engine - Prop Matching Thrust Must Equal Drag Engine Torque Equals Shaft Torque Engine Speeds/Slows to Match Torque Boat Speeds/Slows to Match Thrust Power Reduced If Prop Can't Make Turns J Increase, Torque Increases, Engine Slows Further Major Problems - Wasted Power, Damaged Engines - Particularly With Turbochargers The engine and gear produce torque that spins the shaft. The propeller spins and produces thrust, which overcomes drag to produce speed. Available engine torque depends on the engine, fuel flow, gear ratio, and engine RPM. Required propeller torque depends on RPM, propeller design, and boat speed. Drag depends on hull characteristics and speed. If the engine provides more torque than the propeller absorbs, the shaft speed will try to increase. This will produce more thrust and the boat will speed up. An equilibrium will be achieved between torque, thrust, drag and speed at a higher speed unless the governor on the engine prevents increased RPM. If the RPM can't increase, the boat can't reach the full speed potential of the engine and is "underwheeled". You may not want to accept this condition because it doesn t produce the deseired speed, but it is not harmful. Conversely, if the engine is unable to provide enough torque to turn the propeller, the shaft slows down, and thrust and speed drop, again to equilibrium. However, this "overwheeled" condition won't allow the engine to achieve full RPM and power, so the engine may smoke and lug and eventually suffer damage. This is especially a problem with turbocharged engines because they depend on aiir flow to cool the heads, and in the long run can be damaged by lugging. Note also that we need to know the engine characteristics as well. The amount of either power or torque available varies with RPM. The data for a specific engine is generally available from the manufacturer. However, read the rating conditions carefully for an engine as well. The full power or RPM may not be available for more than a short time, for example. Matching a propeller, gear and engine means that the equilibrium between the available engine torque and the required propeller torque will not overload the engine and that the thrust required to make speed is available throughout the range of operation

20 E Marine Training - Prop Matching - February, Strategy Full Load Vs Part Load Range/Efficiency Vs Top Speed Getting Onto Plane/Top Speed Mixed Service Towing, Trawling, Special Cases Speed Vs Towing Power Economic Analysis Probabilistic Voyage Simulation May Be Required For Fishing or Towing A boat is not generally just required to make top speed on trials day. The owner will generally want a long-lived, fuel efficient match to some specific mission profile (even if he doesn t know it). It is important to fully understand the required mission for a boat before deciding how to prop it. Often range or fuel efficiency may be better with some prop other than the one that gives the highest top speed. A common case for recreational boats is a good hole shot prop, for skiboats. A prop that works best at top end may require everything an engine has to accelerate because it is overloading the engine at lower speeds. The boat thus will accelerate slowly and not be as useful for waterskiing. A prop that is sized differently will have more low speed thrust by letting the engine run up more but won t have as high a top speed. Boats that tow fishing gear or other vessels require some additional thought, because there is generally a conflict between towing performance and top speed. A fishing vessel may need to get to and from the ground quickly, but then tow well. Rescue vessels have to get on site very quickly, and often execute a long difficult tow. Sailing yachts need to balance good speed and possibly range under power with the possible need to power into a heavy blow and to maneuver smartly in confined spaces. This also has to be balanced against the loss of speed under sail. Dinner cruise vessels usually don t need lots of speed, but do need plenty of thrust to dock in a bad blow. In the case of commercial vessels, you need to be ready to do at least simple economic studies, such as how much fuel savings is required to pay off a loan to buy a new prop, and spreadsheet programs offer numerous financial calculation tools to do theses studies. The most sophisticated tool for such studies are probably voyage simulation studies. These are programs that essentially do simple simulations of a voyage, but have randomizers to vary weather, fishing conditions or whatever. They are run hundreds of time to determine the average, best and worse possibilities, and are used to evaluate the economic (or military) effectiveness of just such features as propeller choice. (Or sailing yacht designs in the America s Cup.) The bottom line is that prop matching requires judgment as well as just calculation.